Graphing Linear Equations

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6.3 OBJECTIVES

1. Graph a linear equation by plotting points 2. Graph a linear equation by the intercept method 3. Graph a linear equation by solving the equation for y

We are now ready to combine our work of the previous two sections. In Section 6.1 you learned to write the solutions of equations in two variables as ordered pairs. Then, in Sec- tion 6.2, these ordered pairs were graphed in the plane. Putting these ideas together will let us graph certain equations. Example 1 illustrates this approach.

Graphing a Linear Equation Graph x 2y 4.

Step 1 Find some solutions for x 2y 4. To find solutions, we choose any convenient values for x, say x 0, x 2, and x 4. Given these values for x, we can substitute and then solve for the corresponding value for y. So

If x 0, then y 2, so (0, 2) is a solution.

A handy way to show this information is in a table such as this:

Example 1

NOTEWe are going to find three solutions for the equation. We’ll point out why shortly.

NOTEThe table is just a convenient way to display the information. It is the same as writing (0, 2), (2, 1), and (4, 0).

NOTEThe arrows on the end of the line mean that the line extends indefinitely in either direction.

Step 2 We now graph the solutions found in step 1.

x 2y 4

What pattern do you see? It appears that the three points lie on a straight line, and that is in fact the case.

Step 3 Draw a straight line through the three points graphed in step 2.

y

x x 2y 4

(0, 2) (2, 1)

(4, 0) y

x (0, 2) (2, 1)

(4, 0)

x y

0 2

2 1

4 0

x y

0 2

2 1

4 0

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The line shown is the graph of the equation x 2y 4. It represents all of the ordered pairs that are solutions (an infinite number) for that equation.

Every ordered pair that is a solution will have its graph on this line. Any point on the line will have coordinates that are a solution for the equation.

Note: Why did we suggest finding three solutions in step 1? Two points determine a line, so technically you need only two. The third point that we find is a check to catch any possible errors.

NOTEThe graph is a “picture”

of the solutions for the given equation.

NOTELet x 0, 1, and 2, and substitute to determine the corresponding y values. Again the choices for x are simply convenient. Other values for x would serve the same purpose.

Graph 2x y 6, using the steps shown in Example 1.

y

x

Let’s summarize. An equation that can be written in the form Ax By C

in which A, B, and C are real numbers and A and B cannot both be 0 is called a linear equa- tion in two variables. The graph of this equation is a straight line.

The steps of graphing follow.

Step 1 Find at least three solutions for the equation, and put your results in tabular form.

Step 2 Graph the solutions found in step 1.

Step 3 Draw a straight line through the points determined in step 2 to form the graph of the equation.

Step by Step: To Graph a Linear Equation

Graphing a Linear Equation Graph y 3x.

Step 1 Some solutions are Example 2

C H E C K Y O U R S E L F 1

x y

0 0

1 3

2 6

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Step 2 Graph the points.

Step 3 Draw a line through the points.

y

x y 3x y

x (2, 6)

(1, 3)

(0, 0)

NOTENotice that connecting any two of these points produces the same line.

Graph the equation y 2x after completing the table of values.

y

x

Let’s work through another example of graphing a line from its equation.

Graphing a Linear Equation Example 3

x y

0 1 2

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Step 1 Some solutions are

Graph the equation y 3x 2 after completing the table of values.

y

x

Step 2 Graph the points corresponding to these values.

x y 2x 3 y

y

x (0, 3)

(1, 5) (2, 7)

x y

0 3

1 5

2 7

x y

0 1 2 C H E C K Y O U R S E L F 3

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In graphing equations, particularly when fractions are involved, a careful choice of values for x can simplify the process. Consider Example 4.

Graphing a Linear Equation Graph

As before, we want to find solutions for the given equation by picking convenient values for x. Note that in this case, choosing multiples of 2 will avoid fractional values for y and make the plotting of those solutions much easier. For instance, here we might choose values of

2, 0, and 2 for x.

Step 1 If x 2:

If x 0:

If x 2:

In tabular form, the solutions are

3 2 1 y 3

2 (2) 2

0 2 2 y 3

2 (0) 2

3 2 5 y 3

2 (2) 2 y 3

2 x 2 Example 4

NOTESuppose we do not choose a multiple of 2, say, x 3. Then

is still a valid solution, but we must graph a point with fractional coordinates.

^3,⁵2

5 2

9 2 2 y3

2 (3) 2

x y

2 5

0 2

2 1

Step 2 Graph the points determined above.

y

x (2, 1)

(0, 2) (2, 5)

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x

3

y x 22

y

Graph the equation after completing the table of values.

y

x

y 1 3 x 3

Some special cases of linear equations are illustrated in Examples 5 and 6.

Graphing an Equation That Results in a Vertical Line Graph x 3.

The equation x 3 is equivalent to x 0 y 3. Let’s look at some solutions.

If y 1: If y 4: If y 2:

x 0 1 3 x 0 4 3 x 0(2) 3

x 3 x 3 x 3

In tabular form, Example 5

x y

3 0 3

x y

3 1

3 4

3 2

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What do you observe? The variable x has the value 3, regardless of the value of y. Look at the graph on the following page.

The graph of x 3 is a vertical line crossing the x axis at (3, 0).

Note that graphing (or plotting) points in this case is not really necessary. Simply recognize that the graph of x 3 must be a vertical line (parallel to the y axis) that inter- cepts the x axis at (3, 0).

y

x x 3

(3, 4)

(3, 1)

(3, 2)

Graph the equation x 2.

y

x

Example 6 is a related example involving a horizontal line.

Graphing an Equation That Results in a Horizontal Line Graph y 4.

Because y 4 is equivalent to 0 x y 4, any value for x paired with 4 for y will form a solution. A table of values might be

Example 6

x y

2 4

0 4

2 4

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Here is the graph.

This time the graph is a horizontal line that crosses the y axis at (0, 4). Again the graphing of points is not required. The graph of y 4 must be horizontal (parallel to the x axis) and intercepts the y axis at (0, 4).

y

x (2, 4) (2, 4)

(0, 4)

Graph the equation y 3.

y

x

The following box summarizes our work in the previous two examples:

NOTEWith practice, all this can be done mentally, which is the big advantage of this method.

1. The graph of x a is a vertical line crossing the x axis at (a, 0).

2. The graph of y b is a horizontal line crossing the y axis at (0, b).

Definitions: Vertical and Horizontal Lines

To simplify the graphing of certain linear equations, some students prefer the intercept method of graphing. This method makes use of the fact that the solutions that are easiest to find are those with an x coordinate or a y coordinate of 0. For instance, let’s graph the equation

4x 3y 12

First, let x 0 and solve for y.

4 0 3y 12 3y 12 y 4

So (0, 4) is one solution. Now we let y 0 and solve for x.

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4x 3 0 12 4x 12 x 3

A second solution is (3, 0).

The two points corresponding to these solutions can now be used to graph the equation.

The ordered pair (3, 0) is called the x intercept, and the ordered pair (0, 4) is the y inter- cept of the graph. Using these points to draw the graph gives the name to this method. Let’s look at a second example of graphing by the intercept method.

Using the Intercept Method to Graph a Line Graph 3x 5y 15, using the intercept method.

To find the x intercept, let y 0.

3x 5 0 15 x 5

The x intercept is (5, 0)

To find the y intercept, let x 0.

3 0 5y 15 y 3

The y intercept is (0, 3)

So (5, 0) and (0, 3) are solutions for the equation, and we can use the corresponding points to graph the equation.

3x5y15 (5, 0)

(0, 3)

x y

4x3y12

(0, 4)

(3, 0) y

x

Example 7 NOTERemember, only two

points are needed to graph a line. A third point is used only as a check.

NOTEThe intercepts are the points where the line cuts the x and y axes.

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This all looks quite easy, and for many equations it is. What are the drawbacks? For one, you don’t have a third checkpoint, and it is possible for errors to occur. You can, of course, still find a third point (other than the two intercepts) to be sure your graph is correct. A sec- ond difficulty arises when the x and y intercepts are very close to one another (or are actu- ally the same point—the origin). For instance, if we have the equation

3x 2y 1

the intercepts are and It is hard to draw a line accurately through these intercepts, so choose other solutions farther away from the origin for your points.

Let’s summarize the steps of graphing by the intercept method for appropriate equations.

^0,¹2

^.

¹3, 0

A third method of graphing linear equations involves solving the equation for y. The reason we use this extra step is that it often will make finding solutions for the equation much easier. Let’s look at an example.

Graphing a Linear Equation Graph 2x 3y 6.

Rather than finding solutions for the equation in this form, we solve for y.

2x 3y 6

3y 6 2x Subtract 2x.

Divide by 3.

y 6 2x 3

Graph 4x 5y 20, using the intercept method.

y

x

NOTEFinding the third

“checkpoint” is always a good idea.

Step 1 To find the x intercept: Let y 0, then solve for x.

Step 2 To find the y intercept: Let x 0, then solve for y.

Step 3 Graph the x and y intercepts.

Step 4 Draw a straight line through the intercepts.

Step by Step: Graphing a Line by the Intercept Method C H E C K Y O U R S E L F 7

Example 8

NOTERemember that solving for y means that we want to leave y isolated on the left.

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or

Now find your solutions by picking convenient values for x.

If x 3:

2 2 4

So (3, 4) is a solution.

If x 0:

2

So (0, 2) is a solution.

If x 3:

2 2 0 So (3, 0) is a solution.

We can now plot the points that correspond to these solutions and form the graph of the equation as before.

y

x (3, 4)

(0, 2) (3, 0) 2x 3y 6

y 2 2 3 3 y 2 2

3 0 y 2 2

3 (3) y 2 2

3 x

NOTEAgain, to pick convenient values for x, we suggest you look at the equation carefully. Here, for instance, picking multiples of 3 for x will make the work much easier.

x y

3 4

0 2

3 0

Graph the equation 5x 2y 10. Solve for y to determine solutions.

y

x

x y

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1.

2. 3. 4.

5. 6.

7. 8.

C H E C K Y O U R S E L F A N S W E R S

y

x (3, 0)

( 1, 4) ( 2, 2)

y

x y 2x

y 3x 2 y

x

x 3

3

y 1

y

x

(0, 4)

(5, 0) 4x 5y 20 y

x x 2

y

x

y

x

y 3

x 5

2

y 5 y

x (3, 0)

( 1, 4) ( 2, 2) y

x

6 2x y

x y

1 4

2 2

3 0

x y

0 0

1 2

2 4

x y

0 2

1 1

2 4

x y

3 4

0 3

3 2

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Exercises

Graph each of the following equations.

1. x y 6 ^2. x y 5

3. x y 3 ^4. x y 3

5. 2x y 2 ^6. x 2y 6

6.3

Section Date

ANSWERS 1.

2.

3.

4.

5.

6.

x

y y

x

y

x

y

x

y

x

y

x

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7. 3x y 0 ^8. 3x y 6

9. x 4y 8 ^10. 2x 3y 6

11. y 5x ^12. y 4x

7.

8.

9.

10.

11.

12.

508

y

x

y

x

y

x

y

x

y

x

y

x

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13. y 2x 1 ^14. y 4x 3

15. y 3x 1 ^16. y 3x 3

17. 18. y 1

4 x y 1

3 x

13.

14.

15.

16.

17.

18.

y

x

y

x

y

x

y

x

y

x

y

x

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19. 20.

21. x 5 ^22. y 3

23. y 1 ^24. x 2

y 3 4 x 2 y 2

3 x 3 19.

20.

21.

22.

23.

24.

510

y

x

y

x

y

x

y

x

y

x

y

x

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Graph each of the following equations, using the intercept method.

25. x 2y 4 ^26. 6x y 6

27. 5x 2y 10 ^28. 2x 3y 6

29. 3x 5y 15 ^30. 4x 3y 12

25.

26.

27.

28.

29.

30.

y

x

y

x

y

x

y

x

y

x

y

x

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Graph each of the following equations by first solving for y.

31. x 3y 6 ^32. x 2y 6

33. 3x 4y 12 ^34. 2x 3y 12

35. 5x 4y 20 ^36. 7x 3y 21

31.

32.

33.

34.

35.

36.

512

y

x

y

x

y

x

y

x

y

x

y

x

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Write an equation that describes the following relationships between x and y. Then graph each relationship.

37. y is twice x. 38. y is 3 times x.

39. y is 3 more than x. 40. y is 2 less than x.

41. y is 3 less than 3 times x. 42. y is 4 more than twice x.

37.

38.

39.

40.

41.

42.

y

x

y

x

y

x

y

x

y

x

y

x

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product of 4 and y is 12. y is 6.

Graph each pair of equations on the same axes. Give the coordinates of the point where the lines intersect.

45. x y 4 ^46. x y 3

x y 2 x y 5

47. Graph of winnings. The equation y 0.10x 200 describes the amount of winnings a group earns for collecting plastic jugs in the recycling contest described in exercise 27 at the end of Section 6.2. Sketch the graph of the line on the coordinate system below.

43.

44.

45.

46.

47.

514

y

x

y

x

y

x

y

x

$600

1000

$400

$200

2000 3000 (Pounds)

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48. Minimum values. The contest sponsor will award a prize only if the winning group in the contest collects 100 lb of jugs or more. Use your graph in exercise 47 to determine the minimum prize possible.

49. Fundraising. A high school class wants to raise some money by recycling

newspapers. They decide to rent a truck for a weekend and to collect the newspapers from homes in the neighborhood. The market price for recycled newsprint is currently $11 per ton. The equation y 11x 100 describes the amount of money the class will make, in which y is the amount of money made in dollars, x is the number of tons of newsprint collected, and 100 is the cost in dollars to rent the truck.

48.

49. (a)

(b) (c) (d) 50. (a) (b) (c) (d)

$100

$400

$200

10 20 30 40 50 (Tons)

C

1 2 3 4 5

20 40 60 80

(a) Using the axes below, draw a graph that represents the relationship between newsprint collected and money earned.

(b) The truck is costing the class $100. How many tons of newspapers must the class collect to break even on this project?

(c) If the class members collect 16 tons of newsprint, how much money will they earn?

(d) Six months later the price of newsprint is $17 dollars a ton, and the cost to rent the truck has risen to $125. Write the equation that describes the amount of money the class might make at that time.

50. Production costs. The cost of producing a number of items x is given by C mx b, in which b is the fixed cost and m is the variable cost (the cost of producing one more item).

(a) If the fixed cost is $40 and the variable cost is $10, write the cost equation.

(b) Graph the cost equation.

(c) The revenue generated from the sale of x items is given by R 50x. Graph the revenue equation on the same set of axes as the cost equation.

(d) How many items must be produced for the revenue to equal the cost (the break-even point)?

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Graph each set of equations on the same coordinate system. Do the lines intersect? What are the y intercepts?

51. y 3x ^52. y 2x

y 3x 4 y 2x 3

y 3x 5 y 2x 5

a.

b.

c.

d.

516

Getting Ready for Section 6.4 [Section 1.4]

y

x

y y

x

y

x

y

x

y

x

y

x y

x

Evaluate the following expressions.

(a) (b) (c) (d)

Answers

1. x y 6 ^3. x y 3 ^5. 2x y 2

4 (4) 8 2 4 (2)

6 2

9 5

4 3 7 3

8 4

7. 3x y 0 ^9. x 4y 8 ^11. y 5x

51.

52.

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19. y 2 21. x 5 ^23. y 1

3 x 3

25. x 2y 4 ^27. 5x 2y 10 ^29. 3x 5y 15

31. 33. 35. y 5 5

4 x y 3 3

4 x y 2 x

3

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

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43. x 4y 12 ^45. (3, 1) 47. Graph

49. (a) Graph; (b) 51. The lines do not intersect.

(c) $76; (d) y 17x 125 The y intercepts are (0, 0), (0, 4), and (0, 5).

100

11 or L 9 tons;

y

x

y

x

y

x

y

x

y

x

$600

1000

$400

$200

2000 3000 (Pounds)

$100

$400

$200

10 20 30 40 50 (Tons)

y

x

a. 1 b. 2 c. 3 d. 0

2